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Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces

Published online by Cambridge University Press:  14 June 2011

Y. GUIVARC’H  and
C. R. E. RAJA
Y. GUIVARC’H
Affiliation:
IRMAR, Campus de Beaulieu, Université de Rennes I, 35042 Rennes, France (email: yves.guivarch@univ-rennes1.fr)
C. R. E. RAJA
Affiliation:
Stat-Math Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore 560059, India (email: creraja@isibang.ac.in)
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Abstract

We discuss recurrence and ergodicity properties of random walks and associated skew products for large classes of locally compact groups and homogeneous spaces. In particular, we show that a closed subgroup of a product of finitely many linear groups over local fields supports an adapted recurrent random walk if and only if it has at most quadratic growth. We give also a detailed analysis of ergodicity properties for special classes of random walks on homogeneous spaces and for associated homeomorphisms with infinite invariant measure. The structural properties of closed subgroups of linear groups over local fields and the properties of group actions with respect to certain Radon measures associated with random walks play an important role in the proofs.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012 , pp. 1313 - 1349
Copyright
Copyright © Cambridge University Press 2011

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